Gluon transverse momentum dependent correlators in polarized high energy processes
Daniel Boer, Sabrina Cotogno, Tom van Daal, Piet J. Mulders, Andrea Signori, Yajin Zhou
aa r X i v : . [ h e p - ph ] S e p Gluon transverse momentum dependent correlatorsin polarized high energy processes
Daniel Boer
Van Swinderen Institute, University of Groningen,Nijenborgh 4, NL-9747 AG Groningen, NetherlandsE-mail: [email protected]
Sabrina Cotogno
Nikhef Theory Group and Department of Physics and Astronomy, VU University AmsterdamDe Boelelaan 1081, NL-1081 HV Amsterdam, the NetherlandsE-mail: [email protected]
Tom van Daal
Nikhef Theory Group and Department of Physics and Astronomy, VU University AmsterdamDe Boelelaan 1081, NL-1081 HV Amsterdam, the NetherlandsE-mail: [email protected]
PIET J. MULDERS ∗ Nikhef Theory Group and Department of Physics and Astronomy, VU University AmsterdamDe Boelelaan 1081, NL-1081 HV Amsterdam, the NetherlandsE-mail: [email protected]
Andrea Signori
Nikhef Theory Group and Department of Physics and Astronomy, VU University AmsterdamDe Boelelaan 1081, NL-1081 HV Amsterdam, the NetherlandsE-mail: [email protected]
Yajin Zhou
School of Physics & Key Laboratory of Particle Physics and Particle Irradiation (MOE),Shandong University, Jinan, Shandong 250100, ChinaE-mail: [email protected]
We investigate the gluon transverse momentum dependent correlators as Fourier transform ofmatrix elements of nonlocal operator combinations. At the operator level these correlators includeboth field strength operators and gauge links bridging the nonlocality. In contrast to the collinearPDFs, the gauge links are no longer unique for transverse momentum dependent PDFs (TMDs)and also Wilson loops lead to nontrivial effects. We look at gluon TMDs for unpolarized, vectorand tensor polarized targets. In particular a single Wilson loop operators become important whenone considers the small-x limit of gluon TMDs.
XXIV International Workshop on Deep-Inelastic Scattering and Related Subjects11 - 15 April 2016, DESY, Hamburg ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution 4.0 licence. http://pos.sissa.it/ luon TMD correlators
PIET J. MULDERS
1. Introduction
Parton distribution functions (PDFs) establish the connection between hadrons in initial stateand the hard process. As such they replace in the basic description of the cross section the polar-ization sums for quarks and gluons by correlators , u i ( k ) u j ( k ) (cid:181) / k i j = ⇒ F i j ( k ; P , S ) , (1.1) e a ( k ) e b ∗ ( k ) (cid:181) − g ab T = ⇒ G ab ( k ; P ) (1.2)for a quark with momentum k in a hadron with momentum P . In high energy processes the hadronmomenta defining light-like directions and we use the light-like vector n with P · n =
1. The corre-lators involve quark and gluon fields. The important combination of fields, referred to as leadingtwist , are those minimizing the canonical dimension. We use a Sudakov expansion of the partonmomentum, k = x P + k T + . . . , where . . . is along n and is the irrelevant momentum componentthat can be integrated over leaving correlators F i j ( x , k T ) and G ab ( x , k T ) for quarks and gluons,respectively. Including transverse momenta of the partons they are parametrized in structures withspecific Dirac ( i j ) and Lorentz structure ( ab ) and transverse momentum dependent (TMD) PDFs f ... ( x , k T ) , in short referred to as TMDs.The high energy kinematics is an essential ingredient in this. In the center of mass of thepartonic scattering process, the hadronic momenta are in essence light-like and the hadronic massesbecomes irrelevant. In the hadron rest frame, a given hadron is struck at one particular light-fronttime. In this situation the light-cone fractions x of the parton momentum can be identified withscaling variables, such as the Bjorken scaling variable x B = − q / P · q in deep inelastic scattering.The transverse components of the parton with respect to the hadron can be accessed in processeswhere one observes a non-collinearity in at least three hard momenta, that in the absence of intrinsictransverse momentum in the hadrons are expected to be collinear, of course given a particularpartonic subprocess.This inclusion of transverse momentum and TMD factorization all works straightforward attree-level with factorization theorems being studied. Effects of intrinsic transverse momenta ofpartons are best visible in (partially) polarised processes. In that case one has polarization vectorsor tensors for hadrons (parametrizing the spin density matrix) or measurable polarization vectorsdepending on final state distributions of decay products. The initial state spin (symmetric andtraceless) vectors or tensors for hadrons are in analogy to the momentum expanded as S m = S L P m M + S m T − MS L n m , S mn = " S LL g mn T + S LL P m P n M + S { m LT P n } M + S mn TT − S LL P { m n n } − MS { m LT n n } + M S LL n m n n , ensuring the relations P · S = P m S mn =
2. TMD correlators and distribution functions
The quark and gluon TMD correlators in terms of matrix elements of quark fields [1, 2, 3]2 luon TMD correlators
PIET J. MULDERS including the Wilson lines U needed for color gauge invariance of the TMD case are given by F [ U ] i j ( x , p T ; n ) = Z d x · P d x T ( p ) e ip · x h P , S | y j ( ) U [ , x ] y i ( x ) | P , S i (cid:12)(cid:12) LF , (2.1) G [ U , U ′ ] mn ( x , p T ; n ) = Z d x · P d x T ( p ) e ip · x h P , S | F n m ( ) U [ , x ] F n n ( x ) U ′ [ x , ] | P , S i (cid:12)(cid:12) LF (2.2)(color summation or color tracing implicit). The non-locality in the integration is limited to thelightfront, x · n =
0, indicated with LF. The gauge links U [ , x ] are path ordered exponentials (twodifferent ones for gluons) needed to make the correlator gauge invariant [4, 5, 6]. For the quark cor-relator the gauge link bridges the non-locality, which in the case of TMDs involves also transverseseparation. The simplest ones are the future- and past-pointing staple links U [ ± ][ , x ] (or just [ ± ] ) thatjust connect the points 0 and x via lightcone plus or minus infinity, U [ ± ][ , x ] = U [ n ][ , ± ¥ ] U T [ T , x T ] U [ n ][ ± ¥ , x ] .We use these as our basic building blocks. For gluons TMDs the most general structure involvestwo gauge links (triplet representation), denoted as [ U , U ′ ] , connecting the positions 0 and x indifferent ways. The simplest combinations allowed for [ U , U ′ ] are [+ , +] , [ − , − ] , [+ , − ] and [ − , +] . More complicated possibilities, e.g. with additional (traced) Wilson loops of the form U [ (cid:3) ] = U [+][ , x ] U [ − ][ x , ] = U [+][ , x ] U [ − ] † [ , x ] or its conjugate are allowed as well. A list with all type of con-tributions can be found in Ref. [7, 8]. If U = U ′ one can also use a single gauge link in the octetrepresentation.We note that the combination of two different gauge lines as appearing in the gluon correlatoreven without explicit gluon fields yields an interesting nonvanishing matrix element appearing inthe correlator G [+ , − ] ( x , p T ; n ) ≡ Z d x · P d x T ( p ) e ip · x h P , S | U [+][ , x ] U [ − ][ x , ] | P , S i (cid:12)(cid:12) LF = d ( x ) G [+ , − ] ( p T ; n ) . (2.3)None of the above correlators can be calculated from first principles. They are parametrizedin terms of TMD PDFs, which at the level of leading twist contributions for unpolarized hadrons isgiven by F [ U ] ( x , p T ; n ) = (cid:26) f [ U ] ( x , p T ) + i h ⊥ [ U ] ( x , p T ) / p T M (cid:27) / P , (2.4) G mn [ U , U ′ ] ( x , p T ) = − g mn T f g [ U ] ( x , p T ) + (cid:18) p m T p n T M − g mn T p T M (cid:19) h ⊥ g [ U ] ( x , p T ) . (2.5)For quarks and gluons one can find results in Refs [9, 10, 11, 12], including tensor polarized targetsfor quarks. The gauge link dependence in these parametrizations is contained in the TMDs. Notethat for quarks h ⊥ is T-odd, while for gluons both functions are T-even. At this meeting we reporton the parametrization including tensor polarization in detail outlined in Ref. [13], including alsothe parametrization of the Wilson loop correlator that only depends on p T and for unpolarizedhadrons just is proportional to a scalar function. G [+ , − ] ( p T ) (cid:181) e ( p T ) . (2.6)Even if any gauge link defines a gauge invariant correlator, the relevant gauge links to be usedin a given process just follow from a correct resummation of all diagrams including the exchange3 luon TMD correlators PIET J. MULDERS of any number of A n gluons between the hadronic parts and the hard part, i.e. gluons with theirpolarization along the hadronic momentum. They nicely sum to the path-ordered exponential. Forquark distributions in semi-inclusive deep inelastic scattering they resum into a future-pointinggauge link, in the Drell-Yan process they resum into a past-pointing gauge link, which is directlylinked to the color flow in these processes. Color flow arguments suggest that the Wilson loopcorrelator may be important in diffractive processes.
3. Operator analysis
In the situation of collinear PDFs (integrated over transverse momenta), the non-locality isrestricted to the lightcone, x · n = x T = y ( ) U [ n ][ , x ] y ( x ) | LC or F n m ( ) U [ n ][ , x ] F n n ( x ) U [ n ][ x , ] | LC , expanded in terms of leading twist operators y ( ) D n . . . D n y ( ) and Tr [ F n m D n . . . D n F n n ( ) D n . . . D n ] operators. For transverse momentum dependent correlatorsthe dependence on x T gives in the parametrization distribution functions multiplied with Dirac orLorentz structures involving p T . It is useful to look at the p T dependence in terms of symmetricand traceless tensors (such as for instance in the parametrization of the gluon correlator). The rankof the tensor defines the rank of the distribution function. Rank zero functions are the collinearPDFs.Considering distribution functions of definite rank is also useful because the functions have thesame rank in impact parameter space, which is important for the study of evolution. In principlea factor p T in the parametrization can be rewritten as a derivative working on x T . For distributionfunctions such differentiation gives rise to two types of operators in the correlator, for quarks beingof the form e F [ U ] ˆ O , i j ( x , p T ) = Z d x · P d x T ( p ) e ip · x h P , S | y j ( ) U [ , x ] ˆ O ( x ) y i ( x ) | P , S i (cid:12)(cid:12)(cid:12) LF , (3.1)where the ˆ O ( x ) operators are combinations of i ¶ T ( x ) = iD a T ( x ) − A a T ( x ) and G a ( x ) , defined in acolor gauge invariant way (thus including gauge links), A a T ( x ) = Z ¥ − ¥ d h · P e ( x · P − h · P ) U [ n ][ x , h ] F n a ( h ) U [ n ][ h , x ] , (3.2) G a ( x ) = Z ¥ − ¥ d h · P U [ n ][ x , h ] F n a ( h ) U [ n ][ h , x ] , (3.3)with e ( z ) being the sign function. Note that G a ( x ) = G a ( x T ) does not depend on x · P , implyingin momentum space p · n = p + =
0, hence the name gluonic pole matrix elements [14, 15, 16, 17,18, 19]. The operator in Eq. 3.3 is actually time-reversal odd, giving rise to leading terms in thecorrelators that are important for single spin asymmetries. The factors multiplying the gluonic polecorrelators depend on the gauge links in the correlators, which as already mentioned only dependon the hard process in which the correlators are needed to connect to the hadrons involved. Mostwell-known are the single gluonic pole factors for staple links C [ ± ] G = ± luon TMD correlators PIET J. MULDERS
Depending on the rank of the functions, more gluonic pole operators may enter [20]. Fortwo gluonic poles this lead to an interesting link between the Wilson loop correlator and the gluoncorrelator at x =
0. The Wilson loop correlator only depends on t in the small-x region where k ≈ k T and one finds for x ≈ G [+ , − ] ab ( x , p T ) (cid:181) p a T p b T M G [+ , − ] ( t ) . (3.4)The above results agree with the result in ref. [21] where in the small- x limit f [+ , − ] ( x , p T ) becomesproportional to the dipole cross section. In Ref. [22] that connection was already made on thecorrelator level for the case of a transversely polarized nucleon. Results for tensor polarization areincluded in Ref. [13]. We note that in general combination gluonic poles operators G a with thefield strengths F n a or the quark fields, can give rise to multiple color neutral combinations whichhas to be taken into account.
4. Conclusions
TMDs enrich the partonic structure of hadrons as compared to collinear PDFs. At the technicallevel, there are a number of complications such as the appropriate process-dependent gauge links,the matching of small and large p T and the more complex evolution that need to be addressed. Thestudy of the operator structure of TMDs with definite rank is important and instructive to study therole of TMDs for polarized hadrons and for establishing links to small-x physics. Acknowledgments
This research is part of the research program of the “Stichting voor Fundamenteel Onderzoekder Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Weten-schappelijk Onderzoek (NWO)” and the EU "Ideas" programme QWORK (contract 320389).
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